Dear Antony Locke and list:
Thankyou for the opportunity to clarify what I mean by 'epiphenomenon'.
By the term I meant to convey that the traveling wave, viewed by an outside
observer as a moving wave packet on the CP, doesn't, by itself, DO anything.
It is merely descriptive of the graded time delay in a continuously tuned
bank of resonators that are simultaneously excited, just like the model
Bekesy built where he had a series of pendulums (of graded lengths) hanging
from a common rod that is suddenly jolted. An observer sees 'a traveling
wave' moving along the array of pendulums, but the traveling wave in this
case is an epiphenomenon because it doesn't carry any energy.
On the resonance theory of hearing, what happens is exactly like Bekesy's
model. And so I believe the TW in the cochlea is an epiphenomenon because it
doesn't carry energy from one tuned element to the next.
You state that on your view of cochlear mechanics the TW does carry energy,
and the energy of the TW is shared between the kinetic energy of the fluid
and the potential energy of the CP. I call this the 'flicked rope'
description of cochlear mechanics because it behaves in essence like a
flicked rope, with one element causing the movement of the next (I realise
that you include coupled fluid elements, but the situation is similar in
that one element passes on energy to the next). The arrangement is very
easily modeled by a transmission line, and many mathematical models have
seized on this analogy (and lost sight of what is going on physically). On
this flicked rope description, I would agree that the TW is not
epiphenomenal (it does have causal power), but then I also think this view
is mistaken.
One of the key problems with this flicked rope description is that it relies
on coupling between the resonators to carry the energy. One soon finds that
the coupling is very inefficient, so that the energy becomes prematurely
dissipated way before the wave reaches the CF place. A related problem is
that because one is relying on coupling, each resonating element cannot act
independently, and so it is extremely difficult to achieve the high Q's that
the cochlea does (a spontaneous otoacoustic emission at 1500 Hz can have a
bandwidth of less than 1 Hz (f/deltaf = 1500). I have seen no TW model that
can achieve such a high figure. Even more difficulty arises when one tries
to model a reverse traveling wave (needed in order to carry OAEs from their
location on the partition to the ear canal): the energy losses are even more
severe. Another drawback is that near the threshold of hearing, we are
dealing with minute quanties of energy (units of electron-volts), and how
can the whole partition (at the base) be moved using such tiny energy, even
if the energy is only 'on loan' and is passed on to subsequent CP locations
and eventually to the required CF? Because of these difficulties, one can
see why the OHCs have been called on to pump in energy to overcome these
difficulties, but so far unsuccessfully, I believe.
For these reasons, I have sought to abandon the difficulties posed by
conventional TW theory (defined as that where energy traveling along the CP
is initiated by pressure differences across it) and to develop a true
resonance theory in which outer hair cells sense the common-mode pressure in
the cochlea. At the outset, I want to make it plain that in most practical
respects I am not disputing the general observations of the TW theory: in
particular, there _is_ a graded delay in response along the CP, the only
major difference is that this graded delay is an _epiphenomenon_ (there is
no traveling energy). And so, Antony, I would like you to recognise that I
am not "disregard[ing] two established, and fundamental, observations of
cochlear mechanics."
My resonance theory consists of two components: high level (>60 dB SPL) and
low-level (<60 dB). The low-level component is what my 'Underwater Piano'
paper dwells on, and it calls on the body of the OHCs to respond to fast
intracochlear pressure and in reacting (OHC2 in-phase to pressure; OHC1&3 in
antiphase to pressure) to create a surface acoustic wave (SAW) resonator. I
hope I have described its functioning adequately, and that you can
appreciate that the SAW resonators are tuned from one end of the partition
to the other depending on surface tension (which controls the speed of
ripple propagation in the TM and the resulting resonant frequency of the OHC
cavity). I also hope you can see that the graded tuning will deliver a
graded time delay in response to an acoustic impulse (in accord with
observations conventionally ascribed to a traveling wave). However, the SAW
saturates at about 60 dB when the intracellular potential of all three rows
of OHCs (OHC1, 2, & 3) falls to the same resting potential and there is no
differential gain (OHC2 vs OHC1/3) in the system. Then the high level
component takes over, and here the grading in tuning again relies on the
ripple propagation speed, but in this case the delay comes from the time it
takes for the ripple to propagate from the sharp vestibular lip to the IHCs.
This higher level activity is not described in my Underwater Piano paper,
but its graded delays from base to apex again mimic what happens according
to conventional TW theory. Just to make myself abundantly clear, a
high-level acoustic impulse applied to the stapes will cause fast acoustic
pressure across the tectorial membrane (the speed of sound in the TM is
appreciably different to water) and will cause it to move back and forth at
the place where its mass and compliance are resonant, launching slow ripples
from the vestibular lip. Note that the mass of the TM is continuously graded
from light at the base to heavy at the apex. Clearly, ripples near the base
will travel fastest and reach the nearby IHC first; ripples near the apex
will travel slowest and reach their adjacent IHC last. The result is a
graded delay which _looks_like_ a traveling wave (although no energy is
carried longitudinally; the energy is all absorbed along the length of the
TM during the acoustic impulse and takes various times to travel radially).
Thus, I agree with you that (1) the mechanical properties of the CP are
highly graded, most particularly in the ripple propagation speed of the
tectorial membrane. However, the stiffness and mass of the BM are also
graded, but only so that this absorber is more closely matched to the
characteristics of the TM so that energy can be _absorbed_ (to prevent
damaging overload). Interestingly, you say that the stiffness of the BM
changes by 4 orders of magnitude along the partition, yet according to de
Boer (1980) we need the combined mass and stiffness to vary by 6 orders of
magnitude to tune the CP over three decades, and this appears to be
stretching the data (Phys. Rep. 105, 141-226).
Similarly, I agree with you that (2) it is the fast compressional wave that
sets up a pressure regime in the cochlea which launches more slowly evolving
wave activity. However, the waves are slow ripples on the TM, not a
balancing act between hydraulic elements and the CP.
In practical terms, the main points of divergence I have with classical TW
theory are that:
o at low SPLs: there is no appreciable pressure difference across the
partition. Instead, there is a common-mode pressure (usually thought to have
no effect) which the body of the OHCs are able to detect. Nevertheless, the
active cellular response of the OHCs to intracochlear pressure resembles
that which might happen with a passive differential pressure (and which does
in dead specimens). The OHCs form a tuned SAW resonator via ripples on the
TM, which provides an amplified output to the IHCs. The energy gets to the
OHCs quickly, but it takes time (several cycles) for the SAWs to build up
the energy and pass it to the IHCs. The IHC response is just the same as you
would expect from a TW (but because the _apparent_ TW carries no energy, we
say the TW is epiphenomenal). The common-mode pressure of course is larger
than the differential pressure (perhaps the latter is minuscule, but my
theory doesn't hang on that being the case). If OHCs are pressure detectors,
it follows that they must possess some compressibility. The advantage of
having compressible elements encased within incompressible fluid and rigid
bone is that the sound energy is automatically 'funnelled' to the OHCs via
the fast acoustic wave.
o at high SPLs: the acoustic pressure is developed across the TM, not the
BM. However, the BM ends up vibrating because of the partition is now moving
to and fro, and the BM is there to absorb excess energy.
Thus, in a sense the TW, as commonly understood, is still there on the BM;
it's just that it appears simply by coupling of energy from the principal
resonating elements (on the TM) to the BM. That means that something which
was supposed to be real (i.e. to carry energy via CP/hydraulic interactions)
has suddenly become causally impotent, or epiphenomenal.
I think I have answered your questions, but just to be sure...
>The finding of a highly graded CP, in my opinion, indicates that the
>travelling wave is the mode of energy transfer from base to characteristic
>place; the highly graded CP and travelling wave are intimately linked.
The existence of a highly graded CP is equally compatible with a true
resonance theory or a traveling wave one. That is, the former leads to an
apparent traveling wave (deemed epiphenomenonal); the latter actually
carries energy. Which is correct?
>Andrew and Eckard state that the travelling wave is an epiphenomenon. I
>interpret that to mean that in your model the mode of energy transfer from
>base to characteristic place is independent of the travelling wave.
I can't speak for Eckard, but yes, my position is that the energy gets to CF
by (A) the fast acoustic wave causing activity on the tuned TM which (B)
launches ripples radially toward the IHCs.
>Question 1: Is there any functional relevance of the highly graded CP? If
>so, why is the travelling wave not the obvious candidate for energy
>transfer at the most basic level of description?
The TM is graded in mass so that it resonates against the vestibular lip at
the appropriate place along the partition, the relevant tuning mechanism at
high SPL. The TM's surface tension is also graded to adjust tuning of the
SAW cavity. There is a half-octave disparity in the tuning, leading to the
mysterious 'half-octave shift' of psychophysics. As I said above, the graded
tuning is to be expected in both a resonance theory and a TW theory.
>Question 2: How is the dispersion (i.e. tonotopic: mapping frequency to
>space) property of energy propagation within the cochlea explained by your
>model without invoking the travelling wave?
See longest paragraph above.
I think I've answered your original queries, and I hope you have a clear
understanding of what I'm proposing. If not, ask again. I'm not saying there
is no TW at all, just that the mechanism which underlies its observation in
the cochlea is true resonance, so its appearance is epiphenomenal.
I don't wish to engage in semantic arguments, so if you find the term
'epiphenomenon' unhelpful, I am happy not to use it. I have set out to
construct a simpler, comprehensive description of cochlear mechanics. I
believe my model has the virtue of giving a clear description of which
cochlear elements are resonating and how (it is a 'physically based' model).
It has abundant explanatory power, and can explain things that the
conventional TW theory cannot. It leads to many testable predictions, and I
believe it gives a better account of how we hear.
Thanks for your interest.
Andrew.